Ensem ble- - Based Data Assim ilation Based Data Assim ilation - - PDF document

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Ensem ble- - Based Data Assim ilation Based Data Assim ilation Ensem ble Martin Ehrendorfer Ehrendorfer Martin Department of Meteorology and Department of Meteorology and Data Assimilation Research Centre (DARC) Data Assimilation Research

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Ensem ble Ensem ble-

  • Based Data Assim ilation

Based Data Assim ilation Martin Martin Ehrendorfer Ehrendorfer

Department of Meteorology and Department of Meteorology and Data Assimilation Research Centre (DARC) Data Assimilation Research Centre (DARC) University of Reading University of Reading Workshop on Workshop on Mathematical Advancement in Geophysical Data Assimilation Mathematical Advancement in Geophysical Data Assimilation Banff International Research Station for Banff International Research Station for Mathematical Innovation and Discovery (BIRS) Mathematical Innovation and Discovery (BIRS) 3 3 – – 8 February 2008 8 February 2008 Banff, Alberta, Canada Banff, Alberta, Canada 5 February 5 February 2008 2008

http:/ / w w w .m et.reading.ac.uk/ http:/ / darc.nerc.ac.uk/

Ensemble-Based Data Assimilation, Banff, 5 February 2008 2

  • u t l i n e
  • 1 m otivation

– im portance of DA

  • 2 BT and KF
  • 3 ensem ble-based KF – EnKF
  • 4 specific issues

– degrading xb ps and m odes

  • 5 EnKF variants and directions
  • 6 sum m ary and discussion
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Ensemble-Based Data Assimilation, Banff, 5 February 2008 3

D+10 ECMWF forecast for 05 February 2008 D+9 ECMWF forecast for 05 February 2008

D+1 27/01 D+10 05/02 50m 10m 26/01

216 h

→1-day forecast error different from zero →model has sensitive dependence upon I.C.

10-day & 9-day ECMWF forecast for 05/02/2008/1200UTC

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ERA-4 0 Re-Analysis

Dec – Feb clim atology

7 0 m / s

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Ensemble-Based Data Assimilation, Banff, 5 February 2008 5

The January 0 8 Tellus I ssue

Ensemble-Based Data Assimilation, Banff, 5 February 2008 6

s o m e d e v e l o p m e n t s

  • LeDimet & Talagrand 1986
  • Talagrand & Courtier 1 9 8 7
  • Tokyo 1995 (WMO-II)
  • 4DVAR ECMWF 2 5 Novem ber 1 9 9 7
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Ensemble-Based Data Assimilation, Banff, 5 February 2008 7

adjoint code for grad J

atime : do jt = ntstep , 0 , -1 do k=1,3 ; x(k)=xs(k);xf(k)=xfs(k);xt(k)=xts(k) ;enddo jtp1 = jt+1 ; x (iobs) = x (iobs) + d (jtp1) call adtimstp1 ( x,xf,n,xt,dt,eps,jtp1 ) do k=1,3 ; xs(k)=x(k);xfs(k)=xf(k);xts(k)=xt(k) ;enddo irefc = jt ; call adrhs1t (x,xt,n) do k=1,3 ; xs(k) = xs(k) + x(k) ; xts(k)=0.0 ; enddo enddo atime xs (iobs) = xs (iobs) + d (0)

innovation see fortran printouts grad J

Ensemble-Based Data Assimilation, Banff, 5 February 2008 8

  • u t l i n e
  • 1 m otivation

– im portance of DA

  • 2 BT and KF
  • 3 ensem ble-based KF – EnKF
  • 4 specific issues

– degrading xb ps and m odes

  • 5 EnKF variants and directions
  • 6 sum m ary and discussion
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DA – Bayesian Estim ation Theory B x b DATA y ANALYSI S, Pa x a PRI OR B background error covariance m atrix Pa analysis error covariance m atrix

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data assim ilation in 1 D

analysis background

  • bservation

gain bg error analysis increm ent innovation

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Ensemble-Based Data Assimilation, Banff, 5 February 2008 11

com bination of tw o estim ators recursive filtering

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Kalm an ( 1 9 6 0 ) filter – recursive updating x

covariance update state update gain computation prediction initialization

  • bservation

analysis

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  • u t l i n e
  • 1 m otivation

– im portance of DA

  • 2 BT and KF
  • 3 ensem ble-based KF – EnKF
  • 4 specific issues

– degrading xb ps and m odes

  • 5 EnKF variants and directions
  • 6 sum m ary and discussion

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Pa xa B xb DATA y ANALYSIS, Pa xa PRIOR Ensem ble KF … EnKF – conceptually

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Ensemble-Based Data Assimilation, Banff, 5 February 2008 15

Ensem ble Kalm an Filter - Equations

Analysis Step Forecast Step

+ noise

Gain is formed without ever explicitly forming covariances

Houtekamer & Mitchell 2005 QJ

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EnKF – the analysis step Stochastic update algorithm s – Houtekam er & Mitchell 1 9 9 8 , 2 0 0 5

  • K can be form ed w ithout ever explicitly

com puting the full background B

  • Also: possibility of serial processing

Determ inistic update algorithm s – update in a w ay that generates the sam e analysis error covariances that w ould have been obtained from the KF assum ing the KF‘s B is m odelled from the background ensem ble → square-root filters ( Tippett et al. 2 0 0 3 )

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Ensemble-Based Data Assimilation, Banff, 5 February 2008 17

attractiveness of the EnKF

Fundam entally statistical and physical Com putationally parallel Ability to account for som e aspects of nonlinearity – but fundam entally still least-squares Extendable to treat m odel error Circum vention of current B m odel restrictions Error statistics are flow -dependent and anisotropic Prim ary EnKF issues are treatm ent of sm all ensem ble sizes and treatm ent of m odel error

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attractiveness of the EnKF

Fundam entally statistical and physical Com putationally parallel Ability to account for som e aspects of nonlinearity – but fundam entally still least-squares Extendable to treat m odel error Circum vention of current B m odel restrictions Error statistics are flow -dependent and anisotropic Prim ary EnKF issues are treatm ent of sm all ensem ble sizes and treatm ent of m odel error

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Ensemble-Based Data Assimilation, Banff, 5 February 2008 19

  • u t l i n e
  • 1 m otivation

– im portance of DA

  • 2 BT and KF
  • 3 ensem ble-based KF – EnKF
  • 4 specific issues

– degrading xb ps and m odes

  • 5 EnKF variants and directions
  • 6 sum m ary and discussion

Ensemble-Based Data Assimilation, Banff, 5 February 2008 20

specific issues

I m pact of finite ensem ble size * – w hich com plications w ould disappear if ensem ble size w ere infinite? ( a) Sam pling error/ noise in covariance estim ation * ( b) Covariance localization * ( c) Balance and im balance * ( d) Filter divergence and ensem ble spread * ( e) I nflation, Model error and inbreeding ( use factor slightly > 1 ) ( f) Serial processing of observations ( g) Assim ilation of surface pressure ( h) Use of Lorenz and sim plified m odels ( i) Deficiency in grow th of analysis errors ( j) Ease of im plem entation and costs

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Ensemble-Based Data Assimilation, Banff, 5 February 2008 21

random sam pling: error in ( co) variances is proportional to 1 / N this leads to spurious – long-distance correlations spurious effects of – observations Lorenc 2003 Hamill 2006

noise dom inates ( a) Sam pling noise in estim ated covariances

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( b) covariance localization I m provem ent of noisy covariances – trying to elim inate artifacts

  • f lim ited sam ple size

Schur/ Hadam ard product – A = B o C – I f B and C are covariance m atrices then so is A to act as an heuristic attem pt → – so that distant features do not appear as dynam ically interrelated exam ples: Lorenc, Ham ill, Beck/ Ehrendorfer

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Lorenc 2003 A = B o C

C B A

C … taken from Gaspari/Cohn

sam pled covariances m odified by Schur prd.

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localization exam ple: Ham ill 2 0 0 6

C

A = B o C

B A 2 0 0 m em bers correlation function noisy correlations filtered correlations

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  • perational EPS ( N= 2 5 )

sam pling, N= 2 5 , M= 5 0 sam pling, N= 5 0 , M= 1 0 0 height correlations 5 0 0 hPa derived from ensem ble integrations ( D+ 4 )

spurious correlations removed

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( c) Balance and im balance

Localization causes im balance

– Schur product →

  • height increm ents fall off m ore rapidly w ith distance
  • require increased w ind increm ents for geostrophic balance
  • but w ind increm ents are reduced by Schur product factor

Exam ples:

– Effect of localization on geostrophic balance – significantly subgeostrophic w ind increm ents ( Lorenc 2 0 0 3 ) – PE m odel: im balance less for less severe localization ( Mitchell et al. 2 0 0 2 ) [ m easure ( ?) ]

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( d) Filter divergence and ensem ble spread

if analysis errors are underestim ated in one cycle

– – through sam pling error or neglect of m odel error – – forecast errors m ay be underestim ated in the next cycle, – underw eighting the new observations

feedback: ensem ble assim ilation m ethod progressively ignores

  • bservational data – because background is m odeled poorly –

leading eventually to useless ensem ble know n as filter divergence it is critical to have good background-error covariances

  • Keep ensem ble covariance m atching that of the error

m ethod: covariance localization / inflation / splitting ensem ble filter divergence can also occur in perfect m odel situations,

  • because covariances w ill incur large sam pling errors w hen

estim ated from sm all ensem bles

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filter divergence … the m ajor design problem is to keep the ensem ble covariance m atching that of the error … A Lorenc … one of the m ost crucial preventatives to avoid filter divergence is to m odel background error covariances realistically … T Ham ill

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( e) inflation, m odel error and inbreeding Real w orld: forecasts w ith im perfect m odel

– if – in this situation Q is neglected → – propagated analysis ensem ble has too little spread

Methods to account for this deficiency

– stochastic- dynam ic m odel equations – add noise to each m em ber at the assim ilation tim e – m ultim odel ensem ble to estim ate covariances – covariance inflation

Question:

– is such m odel error realistic? – forecast errors can be greatly out of balance

Question:

  • w hat do w e know about m odel error ?
  • w hat kind of experim ents to assess properties ?

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( e) I nflation ( cont) Ham ill & al 2 0 0 1

„ ...1 % inflation factor optim al in all experim ents, m uch m ore im provem ent is

  • btained through changing correlation length …“
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I nflation ( continued)

Anderson ( 2 0 0 1 ) uses various inflation factors for different variants of EnKF … 1 .0 8 , 1 .1 6 , 1 .0 0 5 , 1 .0 2 , 1 .0 1 See also Evensen ( 2 0 0 3 ) [ review ] Relationship to filter divergence Question:

– Other w orks ? – W hy is such a sm all factor of 1 % of such critical im portance?

Ensemble-Based Data Assimilation, Banff, 5 February 2008 32

  • u t l i n e
  • 1 m otivation

– im portance of DA

  • 2 BT and KF
  • 3 ensem ble-based KF – EnKF
  • 4 specific issues

– degrading xb ps and m odes

  • 5 EnKF variants and directions
  • 6 sum m ary and discussion
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Degrading the background

  • n average ->

the analysis is alw ays m ore accurate than the background

  • n a case-to-case basis ->

how likely is it that the analysis is further from the truth than the background ?

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1 D equations

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analysis and background error

w henever d> 0 then x_ b is degraded w hich is alw ays < 0

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P( d> 0 ) through sam pling

6% E( d) = −k

for example, when, σ_o = 0.1 σ_b 6% chance for d>0 when data are poor

  • > equal chances for

improving/ deteriorating the background 50% for d>0 50% for d<0

poor data good data

poor data: d is negative 50% of the time, implying improvement, but observations are „useless“ … look for more than 50% x_b improvement to conclude

  • bs are valuable

Ehrendorfer 2 0 0 7

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Ensemble-Based Data Assimilation, Banff, 5 February 2008 37

  • u t l i n e
  • 1 m otivation

– im portance of DA

  • 2 BT and KF
  • 3 ensem ble-based KF – EnKF
  • 4 specific issues

– degrading xb ps and m odes

  • 5 EnKF variants and directions
  • 6 sum m ary and discussion

Ensemble-Based Data Assimilation, Banff, 5 February 2008 38

( f) serial processing of observations

Observations can be assim ilated sim ultaneously ( in one batch) or serially ( sequentially, in m any batches) – as long as they have independent errors uncorrelated w ith the background … satisfied to w hat degree ? – then → the result is the sam e I nversion of the scalar ( H B H^ T + R) is trivial The analysed ensem ble from one batch becom es the background ensem ble for the second batch The analysed ensem ble carries m ore and m ore inform ation, but it rem ains a low -dim ensional ( N) subspace

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serial processing

Ehrendorfer 2 0 0 7

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… reconstruct details of the model‘s state at all levels when only p_s

  • bservations are available …

Anderson et al. 2005 JAS Compo et al. 2006 BAMS Whitaker et al. 2004 MWR

HOW ? IMPLICATIONS ? ( g) Assim ilation of p_ s only

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Ensemble-Based Data Assimilation, Banff, 5 February 2008 41

Experim ental setup: a pair of analyses ( 1 ) m odal correction ( blue and red) ( 2 ) reference DA schem e ( dark red) [ for com parison]

(1)

truth generated with AMIC analysis background „OSSE like“

(2)

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correction in NM space m ake analysis perfect for selected NMs

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Ensemble-Based Data Assimilation, Banff, 5 February 2008 43 b1iz transient KE; int. time (days) = 0.969 level 5

mi/ma/rms/x/std 2.3366E+00 2.4767E+02 9.2973E+01 7.6363E+01 5.3036E+01

normalized by 16.5113

1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 10 10 10 11 1 1 11 1 2 12 13 13

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b1ku transient KE; int. time (days) = 250.000 level 5

mi/ma/rms/x/std 2.1875E+00 1.6515E+02 5.0000E+01 3.8852E+01 3.1472E+01

normalized by 11.0100

1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 6 6 6 6 6 6 7 7 7 7 7 7 8 8 8 8 8 9 9 9 9 9 1 10 10 11 1 1 1 2 1 2 1 3

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b1la transient KE; int. time (days) = 500.000 level 4

mi/ma/rms/x/std 6.7212E-01 1.1590E+02 3.5969E+01 2.5812E+01 2.5050E+01

normalized by 7.72667

1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 5 5 5 5 5 5 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 9 9 9 9 9 1 1 10 10 10 1 1 11 11 1 1 12 13

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b1jj transient KE; int. time (days) = 500.000 level 4

mi/ma/rms/x/std 7.6120E-02 1.0940E+02 2.1854E+01 1.3136E+01 1.7465E+01

normalized by 7.29333

1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 5 5 5 5 5 6 6 6 6 6 7 7 7 7 8 8 8 9 9 9 10 1 11 11 1 2 13

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KE of the transient flow com ponent at around 5 0 0 hPa, in units of J/ kg. Upper left: observed ( derived from ECMW F analyses over 2 0 0 0 -2 0 0 5 ) ; Upper right: T1 0 6 L9 ( b1 ku) ; Low er right: T4 5 L6 ( b1la) ; Low er left: T4 5 L6 ( b1 jj) . The good quantitative agreem ent of the results in the right colum n w ith observations is noted. Note also that the use of a relaxation tim e scale of 2 0 days in these experim ents leads to im provem ents, especially in the Southern Hem isphere com pared to the experim ent show n in the low er left using a 1 0 -day relaxation tim e.

215 J/kg

  • bserved

AMI C m odel – KE of transients

85 J/kg 110 J/kg 50 J/kg

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mode 1 mode 2 48 h zcor =1.0 24 h zcor=0.5

13.5 11.5 40 10 110 102 103 103

full correction of mode 2 → error is 10 times larger when compared to mode 1 correction

AE und BG error

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Ensemble-Based Data Assimilation, Banff, 5 February 2008 45

mode 1 mode 2 48 h zcor =1.0 24 h zcor=0.5

6 days 5 days error doubling times in days

but: m ode-1 corrected AE grow s faster

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nonm odality

AE reduced substantially w hen b-m ode rem oved But in term s of forecast error grow th:

– faster w hen using b-m ode corrected analysis … – as com pared to higher m ode corrected analysis – how do rem oved com ponents project on SVs ? …

  • do b-m ode rem oved com ponents have slow projections ?

…

m ore generally: w hich BG error com ponents

– ( m odal vs non-m odal) should DA focus on; – or: to w hich com p‘s is DA restricted to in ExKF ensem ble approach

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Ensemble-Based Data Assimilation, Banff, 5 February 2008 47

Gelaro et al. MWR 2000

black contours: analysis difference (AD) in terms of ζ AD-projection

  • n leading 5 SVs

time-evolved AD-projection * nonm odality

baroclinic → barotropic * rem ove tiny rapidly-grow ing AE portion

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( h) Use of Lorenz m odels in DA

W hat can w e learn from Lorenz m odels?

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Ensemble-Based Data Assimilation, Banff, 5 February 2008 49

( i) Realistic grow th rates

it is critical that m odel has correct error grow th rates m issing sm aller scales – are affecting larger scales through nonlinearity w hite P^ a ( through m ore and m ore observations) – slow grow th – because of large dam ping part of spectrum results m ay in that case be overly optim istic – analysis error too sm all ( danger of filter divergence)

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stationary variances and gain modal amplification

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( j) I m plem entation and costs

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  • u t l i n e
  • 1 m otivation

– im portance of DA

  • 2 BT and KF
  • 3 ensem ble-based KF – EnKF
  • 4 specific issues

– degrading xb ps and m odes

  • 5 EnKF variants and directions
  • 6 sum m ary and discussion
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Ensemble-Based Data Assimilation, Banff, 5 February 2008 53

5 ) EnKF Variants and Directions

EnSRF … Ensem ble square-root filter

– B = X X^ T = ( XU ) ( XU ) ^ T … w here … U U^ T = I

ETKF … Ensem ble Transform Kalm an Filter

– cannot apply covariance localization ( different spaces)

EAKF … Ensem ble Adjustm ent Kalm an Filter

– find adjustm ent m atrix

these are m ethods w ithout perturbed observations

  • determ inistic update algorithm s

square-root filters

  • perform ing SVDs and are thus w ell-behaved

Particle filters LEKF / LETKF Hybrid m ethods, KAEs, balance

Ensemble-Based Data Assimilation, Banff, 5 February 2008 54

Hybrid m ethods based on RRKF ( ?)

Haas 2 0 0 4

blending of static and dynam ic backgrounds

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Ensemble-Based Data Assimilation, Banff, 5 February 2008 55

SI AM New s – the KF and w eather

Fisher 2007

Ensemble-Based Data Assimilation, Banff, 5 February 2008 56

6 ) Sum m ary and discussion

Sam ple size and m odel error Maintain realistic error covariance m odel Cyclic nature of problem Localization – balance – filter divergence – inflation Model error and serial processing Perturbed observations, variants of EnKF I nnovations y – H xb m onitoring Slow grow th of analysis errors Com putational expense and im plem entations

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Ensemble-Based Data Assimilation, Banff, 5 February 2008 57

References

Evensen, 2003: The Ensemble Kalman Filter: theoretical formulation and practical

  • implementation. Ocean Dynamics, 53, 343-367.

Hamill, 2006: Ensemble-based atmospheric data assimilation. In: Predictability of Weather and Climate, Cambridge Univ. Press, pp. 124-156. Lorenc, 2003: The potential of the ensemble Kalman filter for NWP – a comparison with 4D-Var. QJRMS, 129, 3183-3203. Houtekamer and Mitchell, 2005: Ensemble Kalman giltering, QJRMS, 131, 3269-3289. Ehrendorfer, 2007: A review of issues in ensemble-based Kalman filtering. Met. Z., 16, 795-818. Kalnay, 2007: 4dvar or ensemble Kalman filter? Tellus, 59A, 758-773.

…

Ensemble-Based Data Assimilation, Banff, 5 February 2008 58